What is Mathematics for STEM? What is a Mathematics of Solution?

Why all the Fuss about Multiplication

        We only consider multiplication of non-negative integers. (multiplication in the natural                        number system). When we say "multiplication" the operands and the product are natural                      numbers.

I think it's because multiplication is the first encounter with arithmetic that is more than just elaborated counting. Students do not bring applicable intuition: multiplication must be taught and learned in an academic setting. The traditional approach skirts its difficulty one, or both, of two ways:

• it says multiplication is (multiple) addition so there is really nothing new, and/or
• it says its too difficult to understand it, remember the multiplication tables.

I assume each of these approaches contributes to an efficient route to worthy goals, but their pursuit short circuits the opportunity to see what multiplication really is. In the past it could be argued that students only had a need for vocational math, hence it wasn't helpful to know how arithmetic worked; it was just a tool, learn how to do it fast. That was probably a reasonable argument 50 years ago. It's a lousy argument for the world where today's student will work. Understanding how things work is the way to be effective with and around computers/processors, and that is where today's students will be. One doesn't compute with computers, one tells computers how to compute. To do that you need to know how computation works

Why is it Hard to Understand Multiplication?

I think there is a simple explanation: the definition of multiplication involves multiplication itself; it also involves addition. We all know what is being defined can not also appear in its definition; but it turns out there is an escape clause. We are interested in the definition of the multiplication operation, and, although the multiplication operation does appear in its own definition, one of its operands is different, and that allows the operation to work itself out as a process. This is best seen in an example. Consider 6 x 4 with the natural number multiplication definition:  

• 6 x 4 = 6 x (3 + 1) + 6 = 6 x 3 + 6 =

Read this as "(six times four) equals/is  (six times three) plus six". The multiplication operation being defined does appear on both sides of the equals sign, but what happens when we substitute the same definition for 6 x 3, and continue in this manner:

• 6 x 4  =

1. 6 x (3+1) = 6 x 3  +  6 x 1 =  6 x 3 + 6
2. 6 x (2+1) + 6 = (6 x 2 + 6 x 1)  + 6 = (6 x 2  + 6) + 6
3. (6 x (1+1)) +6)  + 6 = ((6 x 1 + 6 x 1) + 6) + 6 =   (( 6 + 6 ) + 6 ) + 6
4. (12 + 6) + 6 = 
5. 18 + 6 = 

• 24

The definition specifies a process for multiplication. When you expand the definition for an actual multiplication, you write down a process:

• multiplication is a process. 

In fact it is a concatenation of two processes, one for multiplication and the other for addition. Each step in a multiplication(addition) process is followed by another multiplication(addition) step until the process terminates; each of these processes is recursive. 

And finally we have spoken the name of a process that refers to itself.  The idea of a recursive process can be hard to grasp; the best way to get a feeling for recursion is to do recursions. Laying out the process as the one above is, reveals its dynamic as a pattern: it first lays out the decomposition of the multiplication one sub-multiplication at a time and then it collects these terms into one sum by one addition at a time. It first goes out and then it comes back:
      multiplication is an process executed as

• a recursive multiplication process followed by
• a recursive addition process. 

A two step process is a general architecture for problem solution:

1. Formulate an expression for the solution that can be evaluated.
2. Evaluate the expression.

The first step is often neglected in school mathematics. Multiplication is an example of that neglect; it is often defined as the multiple addition produced by the first step, and then it is said "multiplication is multiple addition". This approach takes for the solution an expression where the results of the multiplication in the first step have already been incorporated; it is easy to overlook in natural number multiplication where n x 1 = n. In word problems, problems that involve quantities in the real world, the product is not the same thing as either of the operands: the product is not just a different number of what an operand is for natural numbers.

And Your Point Is?

Its slogan could be "multiplication is multiplication plus addition":

• Multiplication is a process, 
• the process terminates when it produces a product.
• To understand multiplication is to understand that process.
• It is a recursive process:"multiplication is multiplication plus addition" 

Multiplication as a solution is dynamic. It is not just the inert answer to a problem. In this, it is an exemplar of the solutions that productive members of society construct. 

For the most part school mathematics is now taught as if the problems of mathematics education itself had been solved. Success is just figuring out how to get students, teachers, and the rest of us to buy in to a program that says: we lay it all out for you, all you have to do is retrieve it. It is a long way from here to "the use of this stuff is in the understanding of it". I want to participate in that journey. 

Values of One

Numbers Keep the Score. They Don't Play the Game.

I started this blog to force myself to look for the reason(s) I found it difficult to do problems in freshman university physics. Many of the friends I talked to experienced the same thing when they first confronted problems with their school mathematics. It does appear to be a common problem in US math education: students are fine with straight forward arithmetic, but have a big problem seeing how it is applied.

• one of the greatest deficits in U.S. students learning mathematics is in their ability to solve problems( page 138 in Adding it Up, National Research Council, 2001). 

I think I can finally see what could cause the problem, and I even think there is a practical way to address it: little is offered in school arithmetic to help a student focus attention on the problem itself, to first think what the problem is about; what objects are involved? how are they related? and from this perspective, work toward an expression of the problem. In my experience and reading, school mathematics is taught the other way around: the numbers drive the strategy. If this is the perspective taken in formulating problems, it could be a reason they often feel contrived, more like puzzles than problems.

"Expression" can suggest it is necessary to wait for students' exposure to school algebra before they have the tool to express problems. This is the conundrum: until algebra there appears to be no way to express problems, but students get to algebra already expecting numbers to guide them to solution of word problems. Before students get to algebra they need a low barrier approach to learning what the objects in a problem are, and how they act and how they are related.

To be a successful word problem solver a student needs to able to call upon a conceptual framework for solving word problems. Such a framework is not a memory file of formulas. It is the concept structure built from knowing that words contain the problem:

• words convey the information that must be understood in order to formulate the problem;
• a general formulation of the problem exists that does not depend on the numbers.

Successful problem solvers must conceive of themselves as solution formulators, not arithmetic evaluators. To do that, a student needs a concept that connects a problem's expression to the construction of the expression of its solution.  

Values of One

School mathematics problems concern quantities resulting from making a measurement or from counting, actually or conceptually. In either case the quantity is expressed in "units", where a unit is one of something. That "something" is the "value of one". It determines how the quantity for which it is the unit participates in the expression of the solution to the word problem. The numerical value of a quantity expresses the magnitude, not the manner, of its problem participation.

Ignoring values of one, what we will generally call "unit values" from now on, has resulted in misunderstanding the multiplication introduced in early elementary education. The product of two natural numbers is a natural number; multiplication that happens in the natural number system stays in the natural number system. Within the natural number system the unity "1" is itself the unit value, and for any natural number n, n x 1 = 1.

A number system whose unit value is not simply "1", say (1 rabbit) or (1 foot), does not have a unity value. The product (1 rabbit) x (1 rabbit) is not in the same number system as (1 rabbit) {or should we say set/collection}. Addition can only be performed within the same number system, and this holds for both natural numbers and and non trivial unit values

Examples of Unit Values

For an object to participate in a analytic problem it must have its own unit, which serves to state the terms of its participation: are the apples participating as

• individual apples with a unit of 1-apple?
• as individual pieces of fruit with a unit of 1-fruit?
• as a collection of 12 apples with a unit of 1-doz-apples?
• as a pound of apples with unit of 1-lb-apples?
• as a fruit in a collection of apples and oranges?

Is time participating as a 

• 1 second,
• 1 week
• 1 average-squirrel-lifetime 

Is distance participating as 

• 1 mile,
• 1 mm

Are rabbit cages participating as 

• 1 rabbit-cage
• 1 doz rabbit-cages  

And for rabbits

• 1 rabbit
• 1 pair of rabbits
• 1 buck rabbit
• 1 doe rabbit

Numbers for counted quantities generally behave numerically as whole/natural numbers; units for measured quantities, as real numbers. Measured quantities are associated with a continuum; a number for a measurement is often expressed as an integer value embedded in a continuum . More generally we can assume measurements are expressed as rational numbers embedded in that continuum; this makes it possible to formulate measurement problems amenable to school arithmetic without worrying too much about number systems.

Arithmetic and Units

* I assume problems addressed to the elementary level can only rely on evaluation of an expression, not on the algebraic manipulation of variables across an equal sign.

A major difference between addition and multiplication is easily expressed in terms of the unit value of their operands:

• Addition: there is only one unit value; each operand, as well as the sum have the same unit valuee.
• Multiplication: there can be three distinct unit values, and there generally there are. 

This difference does not arise when you are simply multiplying two natural numbers where it can appear that multiplication is a variant of addition. If addition were the only available operation in arithmetic, the universe of addressable problems would be small indeed: addition can only deal with augmentation by a like quantity. Its single unit value factors out of the addition expression, and hence there is essentially only one type/kind of arithmetic addition problem, and it makes sense to teach/drill natural number addition without reference to a specific problem. If you conclude addition is the operation for a problem, then you will have identified the unit value, and can proceed to execute the numerical addition and then append the unit to the sum.

If you do not see augmentation as the problem, then you are facing different issues: what are the objects? their unit values? and how do the unit values fit together to express the problem? 

Multiplication Problems

The post before this one considered natural numbers with unit values. In fact it derived the arithmetic for them. The work there shows how the difficulty students now have with word problems creeps into their education at its earliest stage: they are taught the basis of multiplication is a table of small-natural-number results that can stand alone with no consideration of unit values. From the very beginning, mathematics is presented as separable from the problems it addresses; that it is somehow self contained within the numbers it considers.   Instead of telling students "multiplication is xxx", should not we be putting ourselves in position to say "the multiplication we do works like this: xxx"?

First some notation anguish; what would better serve young learners? Something that carries the notion of context beyond the usual "3 feet". Let us try "3:ftW" for 3 feet of width. It is not standard, but it better expresses the close coupling of number to unit value than the standard stand-off notation; for a given length, the number would not be 3 unless it referred to feet.

An Area Calculation 

Let's review some natural number work. With the notation, "[ ]" for successor introduced in the last post we have

• 1:ftW = [0:ftW] = 0:ftW + 1:ftW,
•  
•  
• 4:ftW  =  [3:ftW]  =  3:ftW + 1:ftW

indicating 1:ftW is the unit for Natural Width Numbers. Now we can construct and evaluate the expression for a 6 foot by 3 foot rectangle:

• 6:ftL x 3:ftW  =  6:ftL x [[[0:ftW]]] = 
• 6:ftL x ( 2:ftW + 1ft:W)  =  6:ftL x 2:ftW + 6:ftL x 1:ftW   =
• (6:ftL x 1:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW  = 
• ((6:ftL x 0:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW) + 6:ftL x 1:ftW)  =
• ((6:ftL x 1:ftW + 6:ftL x 1:ftW) + 6:ftL x 1:ftW) 

The multiplication is now the sum of  3 terms, each the product 6:ftL x 1:ftW; the multiplication of length by width is now the sum of the area of three sub rectangles:

• 6:ftL x 1:ftW = (6 x 1):(1:ftL x 1:ftW) = (6 x 1) x 1:ftftA = 6:ftftA,

where the product of unit values is the unit value of the unit value product: 1:ftftA for "feet squared" or "square feet". We resume the evaluation:

• (( 6:ftL x 1:ftW + 6:ftL x 1:ftW ) + 6:ftL x 1:ftW ) = 
• (( 6:ftftA + 6:ftftA ) + 6:ftftA ) =
• ( 12:ftftA + 6:ftftA ) =
• 18:ftftA 

The full decomposition is provided to show how the multiplication expression is first expanded into a sum of terms from the decomposed product, in this case three  6:ftL x 1:ftW sub areas each of area 6 square feet. 

This is the way of multiplication: the multiplication itself is sequentially reduced by taking one unit value from one of the multiplicands in the multiplication to make a "1-unit-value multiplication by the other multiplicand" contribution to the product. Multiplication includes recursion process, not just the final assembly of the sum, and it is necessary for it to do so in order to obtain an expression that only involves multiple multiplications by 1 unit value; all we know coming into this calculation is multiplication by one unit value; this multiplication is constructed, not retrieved from memory. 

This is an instance of confusion arising from conclusions drawn from the multiplication of unitless Natural Numbers. There we have,

• n x 0 = 0 and n x [m] = n x m + n,    

If we take that over into unit natural number multiplication, we would be led to the error: 

• 6:ftL x 3:ftW = (6:ftL + 6:ftL) + 6:ftL.

For the natural number n, n x 1 = n, while 6:ftL x 1:ftW = 6:ftftA. Unit values matter. In fact the solution to the problem follows from understanding area and its relation to the perpendicular sides of the rectangle, and that understanding is contained in the unit value relation, 1:ftL x 1:ftW = 1:ftftA, of the product of unit values for the sides and the unit value of the resulting square's area.

Word problems encountered in school mathematics often appear to be reverse engineered from arithmetic problems. For an area calculation like the one above, an academic word problem is often stated as if the point is "6 x 3 =18".  In real word problem solving, the battle has been won by the time you get to the arithmetic. Attending to unit values focuses attention on the factors important to problem solving.

The unit values in the area problem are unit values of length measurement: unit values of a measurement of a physical continuum. They are real numbers, in this case numbers with integer values embedded in a continuum. For counting problem, unit values are equally significant. We will return to some examples in the next section. 

Ratios and Rates in Multiplication

Understanding rates and ratios provides access to a large fraction of the interesting problems involving multiplication. 

A ratio is a comparison of two quantities in a multiplicative context just as a difference is a comparison of two quantities in a additive context. In either case the significance of the quantities is determined by their unit values. Multiplication is the interesting case.

The distance and duration of a trip can be related by their ratio, say 60:mileDis/2:hourDur. This is equivalent to the average speed 30:mile/1:hour = 30:mphSpeed. At this average speed, a 6:hour trip covers

• 30:mphSpeed x 6:hour = (30:mile/1:hour) x 6:hour = 30:mile x (6:hour/1:hour) = 
• (6 x 30):mile = 180:mile.

We now consider a problem based on counted quantities. A rabbit breeder has 18 rabbits and 6 cages. Take a single-rabbit unit value "oneR", and "oneC" for a single cage as the cage unit value, then the rabbit-to-cage ratio is

• 18:oneR/6:oneC = (18/6) x (1:oneR/1:onrC) = 3:oneR/1:oneC.

which says the uniform rabbit cage occupancy rate is 3 rabbits per cage.

Rates can be useful in decomposing a problem where the issue is not obviously a rate. A teacher has 3 books and would like to know how many ways they could be paired with 6 students. Now 1:book can be paired with any one of 6:student: the book pairing rate is 6:B-S per book. The total number of book-student pairings is this rate times the number of books:

• ( 6:B-S /1:book ) x 3:book  =   ( 6 x 3):B-S x (1:book/1:book) = 18:B-S

We return to the rectangle area problem to see the rate at which area accumulates. It is slight variation on the accumulation of chunks of area previously discussed. The rate of area increase per unit width for the 3 foot wide, 18 square ft area follows from

• area per width unit value ratio  =  1:ftftA/1:ftW

for the rectangle under consideration as 

• 18:ftftA/3:ftW  =  (18/3):ftftA/1:ftW  =  6:ftftA/1:ftW.

This says the area increases by 6 square feet per 1 foot of width, which is the length 6:ftL, providing an interesting variation of the rectangle area problem

The point here is: a problem can be addressed through an expression involving only unit values, a general expression for the common kernel in a collection of problems that differ only in their number values.

The Significance of Unit Values in School Mathematics  

For the most part today's students are asked to approach word problems though numerical values and remembered arithmetic operations. They are not provided with any conceptual framework to address the problem on its own terms. They are often left with the idea that each word problem is special unto itself, a creature of the numbers in it, each problem requiring its own sorcery.

Unit values and their relations are the bricks and mortar of a conceptual framework to organize the terms of a word problem, and those terms are stated in words. The words alone describe a general problem, and that is the problem addressed by unit values in what be called a unit expression.

I maintain there are even larger issues at stake here. Now students are assigned an essentially passive role in their mathematics education. They see a mathematics that is essentially a definitional subject where their learning role is largely passive. Mathematics is presented to them as a corpse they are invited to pick at, whose parts they are required to remember. Elementary school students are invited to learn mathematics the way medical students learn anatomy. By way of an example:

• a number line is used to demonstrate numbers are already "there" to be laid out for inspection,
• multiplication is what is in multiplication tables,
• for larger numbers multiplication is what an ancient algorithm does.

Unit values support the construction of a solution to word problems. Students are introduced to a tool, and thereby given agency. They construct a solution to a word problem. A previous post provided a way to construct numbers, as well as addition and multiplication. Constructive mathematics addresses students as potential producers, not as mere consumers, and this is true on several levels. The realistic hope for STEM in elementary school is students will be given the opportunity to see themselves as producers. 

Is this practical? Can it be taught? I see nothing in a constructive approach that is intrinsically more difficult than the current approach for either students or teachers. On the other hand it differs greatly from what a deeply embedded system currently offers. A lot of discussion would need to take place, and much work anticipated to complete a story that speaks to the concerns of educators

  

Constructive Elementary Arithmetic

Need More Learning-to-Understand in School Mathematics

For all intents and purposes, the basis for early school arithmetic is memorized small number addition and multiplication. This can be an efficient way to learn an arithmetic for calculation. It also may have a significant downside: (1) it sends a message that it is good enough to know how to do without understanding, and (2) it forgoes the opportunity to start learning how to understand. Sharply focused tools address immediate problems; learning to understand is the base on which to move forward.

To some students mathematics education can be a series of ever higher barriers, and they often appear to come as surprises, as if the student is seeing something for the first time. It should not be this way. Mathematics at any level can reveal modes of thought and understanding that lower barriers to next level.

 

This post intends to show how to understand early arithmetic with a constructive approach. The slogan might be: arithmetic is built, not memorized. It is based on the assertion that speed is not of the essence, understanding is necessary to get you further along. 

Constructive mathematics is close to the programming of arithmetic. It is much like writing in a computer language. I understand this might be a good thing now.

Of course there is nothing new here. It is the result of trying to follow up simple questions about a foundation for school arithmetic. 

Constructing Natural Numbers

A Natural Number is one of the non-negative integers, the numbers with which we all started our mathematics education. Historically they were considered as "natural" and hence without need of further explanation. This view changed in the 19th century when concerns were raised about the foundations for arithmetic and Euclidean geometry. For arithmetic this eventually lead to the understanding of calculation which plays an important foundation role in Computer Science. Foundation work for the natural numbers provided a simple way to construct them, and this leads to simple construction of arithmetic with them. I think it could be a practical way for beginning students to understand their arithmetic; and to begin learning how to understand mathematics.

        (mostly from Halmos, Naive Set Theory)
Define a natural number as a set/collection that contains exactly that number of elements/units. By analogy think of a meter as being defined by a selected object, that on intuitive or practical grounds deserves to be called one meter long. Then "one-meter-ness"  is attributed to another object just in case it has the same length as the selected object.   Our number is such an object for number-ness; e.g. for 2 we take a set/collection with exactly 2 elements, or a set/collection with
2-number-ness. Where do we get an element with number-ness? We need a scheme that creates elements as we proceed in the construction of numbers, a process that starts with a small seed and then proceeds on its own with a simple repetitive process, a bootstrap with a small seed that brings itself into being.

Let's say we already have a number, say 7 defined as a set/collection with 7 elements. Clearly the number 7 itself can not be one of those 7 elements, so we can take it as the eighth element. We can think of the number 7 as the name of the set that defines it.

Now we are in position to specify the step in the repetitive process that constructs all the natural numbers. Before any step the process has at its disposal all the names of all the sets/collections produced up to this step. For any set n the next or successor set [n] is given by

            [n] = n U {n} ,

where U specifies the union of n, the set/collection of the elements of n; and the singleton set {n} whose element is the name of n. That is every natural number is to be the set of (the names of) its predecessors. This is more straight-forward than it might at first appear. It is easiest to see in the construction of natural numbers shown below.

0 is set/collection with no elements, the empty set/collection:

             0 = { } .

For each number to be the set of its predecessors:

            1 = [0] = 0 U {0} = { } U {0} = {0},
            2 = [1] = 1 U {1} = {0} U {1} = {0,1} ,
            3 = [2] = 2 U {2} = {0,1} U {2} = {0,1,2}  ,
            etc.
Here etc. has its usual meaning: you get the drift, and the process can now be left on its own to produce any natural number we should need. "[-]" is the successor of the number between the brackets. With this construction in mind, it is easy to picture the

Axiom of Infinity: There is a set containing 0 and  containing the successor of each of its elements.

Where did those numerals 1, 2, 3, ... come from? We used the familiar ones, but we could have made up our own; all a numeral does is name a set/set/collection that is filled with names we supplied previously. The problem with custom names: this scheme is designed to construct natural numbers that do what the natural numbers did before the scheme was invented. Mathematics did not invent them, but it did make this scheme to build them so they can be trusted because they can be understood.

All this is elegant and efficient; we have a process to construct all of the natural numbers in terms of the natural numbers themselves that is ordered to generate natural numbers before they are needed to generate succeeding natural numbers. On the other hand it seems like a little too much; we never really use all all this structure for much beyond constructing the numbers themselves. It is like scaffolding: it allows us to construct what we need, but then what it allowed us to build can provide what it was designed for on its own. The good news: you can construct the numbers you will use yourself, you understand how they work, and you no longer have to fall back on the authority of "naturality". Just remember what 0 is, and what it means for each natural number to have a successor:

• 0
• 1 = [0]
• 2 = [1] = [[ 0 ]]
• 3 = [2] = [[[ 0 ]]]
• 4 = [3] = [[[[ 0 ]]]] 
•  
• 
  
•  

Defining Constructive Arithmetic on Natural Numbers

We have numbers, how we do arithmetic with them? What is addition, and what is multiplication, with Natural Numbers? Here's the plan:

• Define addition on Natural Numbers. Then show [] and "+1" have the expected relation [n] = n+1 : we can count to construct a natural number.  
• Introduce the "unit" for the element being counted; i.e,  for the unit "ball",                                .
•  Given a unit, we can construct a number of units, or count out the "number of units"; e.g.          4 balls. A "number of units" is constructed in the same way a natural number is constructed.
• We show: a "number of units" can only be added to a "number of the same units" .

The definitions for addition and multiplication found in the Natural Number Wiki appear to be the common ones. Addition must be defined before multiplication; addition is contained in multiplication's definition.  

Definition of addition,  "+" , in  n + m

There are two clauses:

• n = n + 0,
• n + [m] = [n + m] = [n] + m 

The second clause states the equivalence of the successor of different combinations of the operands:

• n + ( the successor of m) is the same as the successor of ( n + m) is the same as (successor of n) + m. 

This will lead to a strong uniformity constraint on, and among, operands and sum; this will come later.

With 1 = [0] : 

• n + 1 = n + [0] = [n + 0]       using the second clause.
• [n + 0] = [n]                          using the first clause, giving
• n + 1 = [n]                             for any n, its successor is equivalent to n + 1.

With addition we can give a second construction for natural numbers:

•  0
•  0 + 1 = [0] = 1,
•  1 + 1 = [1] = 2,
•  2 + 1 = [2] = 3,
• 
  

 Let's look at the pattern of an addition example working itself out. But first we consider just a single number working itself in: 3 = [2] = [[1]] = [[[0]]], showing the expression of a natural number as 0 clothed in the number of its successors that take it back to, in this case, 3. Since 0 is not the successor of any natural number it it not possible to go further. Consider the the addition: 9 + 3:

• 9 + 3 = 
• 9 + [ [ [ 0 ] ] ] =  
• [ 9 + [ [ 0 ] ] ] =                 using the second clause in the definition of addition 
• [ [ 9 + [ 0 ] ] ] =                 and again
• [ [ [ 9 + 0 ] ] ] =                 and yet again, and for the third time.
•    [ [ [ 9 ] ] ]    =                     using the first clause ) 
•     [ [ 10 ] ]      =  9 + 1                     10 is the successor of 9 ,
•       [ 11 ]         = (9 + 1) + 1             11 of 10.
•          12           = ( (9 + 1) + 1) + 1 = (10 + 1) + 1 = 11 + 1 

If we are counting a set/collection of "these" where "these" could be balls, days, elephants, nails, ... then 0 of "these" is an empty set/collection of and for "these", and the successor of 0, [0] is 1 of "these". Let's take ball for 1 of "these". We can now construct the set of ball enumerators:

• 0 ball
• [0 ball] = 0 ball + 1 ball = 1 ball,
• [1 ball] = 1 ball + 1 ball = 2 ball,
• [2 ball] = 2 ball + 1 ball = 3 ball,
• [3 ball] = 3 ball + 1 ball = 4 ball.
•  
•  

4 ball  =  ((((0 ball + 1 ball) + 1 ball ) + 1 ball) + 1 ball) and etc. All those parenthesis are there because we are describing a process, not just describing a relationship, and we only know how to add two numbers. In any case 4 ball can be constructed by 4 consecutive additions of 1 ball to 0 ball.

4 ball can also be constructed by 4 successor steps of 0 ball. 

• [[[[ 0 ball ]]]] = 
• [[[ 1 ball ]]]] = 
• [[ 2 ball  ]] = 
• [ 3 ball ] = 
• 4 ball 

And now we have to confront a basic question: what does "4 ball" mean? . We construct "4 ball" by four successor steps from "0 ball" ( we know this is equivalent to four consecutive additions of "1 ball" ). The sense of "4 ball" is it is the thing we constructed in four steps, each step incrementing by "1 ball"the count that started with 0. If we think of "Ball" as the name of a set/collection of balls from which one ball is removed and added to the set of counted balls, just as the above process does, until the ball set/collection is empty, then by doing that process we will have proved, by construction, there were 4 balls in Ball. In this case "4 ball" is a sentence, something like" there are four balls in Ball", and the above process is a proof this sentence is true. And the purpose of mathematics is to write true sentences about significant mathematical objects like numbers, triangles, a space, ... We will return to the notation for a "number with unit" after we have defined natural number multiplication.

An elementary school student knows or learns, in an informal way, this proof and it is the sense of this proof that is the student's earliest number sense. It is their number sense before they learn arithmetic, but when they turn to arithmetic a strange thing happens, or so it seems to me. Instead of building out their intuitive, informal understanding of constructing numbers in order to have the foundation for a constructive arithmetic, school education turns to an arithmetic of natural numbers that is contained in tables that are to be memorized. This is all well, if not good, if the student is just being prepared for speed with simple arithmetic with numbers, but this is not the case; students are expected to apply their arithmetic to problems beyond numbers themselves, to word problems that can ask ask for multiplication of rabbits by cages.

A rabbit and a cage are examples of units, just as ball was.   

Multiplication: n x m

for n and m any Natural mumber. Again there are two clauses in the definition:

• n x 0 = 0,
• n x [m] = n x (m + 1 ) = n x m + n.

Note, the multiplication definition uses addition and the relation between addition of 1 and successor: [n] = n + 1. We also note that n x 1 = n x [0] = n x 0 + n = n. This is expected, but for multiplication of numbers with units will have to be reconsidered.

Here is an example of multiplication of natural numbers:

          9  x 3   =

• 9 x [ [ [ 0 ] ] ] =                          (using clause 2 and [m] = m + 1)           
• 9 x ( [ [ 0 ] ] + 1 )  =  9 x [ [ 0 ] ] + 9   = 
• (9 x [0] + 9) + 9   =  ( (9 x 0 + 9 )  + 9 )  +  9   = 
• 
  
• ( (9 + 9) + 9) =
• ( 18 )  +  9    =           
• 27 .   

This example again shows the expansion of the expression as the multiplication is layed out, followed by contraction as the expression is evaluated by one addition at a time. A multiplication is both expansion and contraction, it is both "a taking apart" and a "putting together". The taking apart uses step-wise dis-assembly of one operand to produce a multiple addition that is evaluated, as it must be, one addition at a time. What else could it involve? To make multiplication we only have what we have already constructed, or at least that is all we have for defining multiplication, and multiplication is defined. 

Unit and Multiplication

We return to the question of notation; what does "1 ball" mean? OK it means "one ball", but then we say "2 ball" is its successor. It feels like we are thinking of it as one word, in fact we are, so from now on we use "1_ball for "1 ball" and "n_ball" for "n ball". Remember numbers can only refer to one "thing"; "one thing" can be "one thing or another thing", but that really counts as "one thing". The two operands and the sum in addition are numbers for the same "thing".

Multiplication is altogether a different matter. The numbers in each operand and the product can be numbers of different "things". For multiplication we must learn how to work with this difference. An addition problem deals with one "thing", and the arithmetic for addition could be considered independently of that "thing". This is the not the case for multiplication.

We need to conform with the common nomenclature: "1_ball" is the counting unit for balls, and etc. To to do useful work with multiplication a student needs unit sense, number sense is insufficient. The secret for solving real problems is to get the units straightened out first, the operations second, and the numbers last.

There are Three Different Multiplications ,

1. Only one operand has a unit. These are multiplications where "x" is often pronounced "times". Examples:                                                                                                                       ** An example is: n x (m_cages) =  m x (n_cages) = (n x m)_cages. These can be derived by arguments you must be tired of.                                                                                                     ** 3 x (6_rabbit) = 3_rabbit x 6 = (3 x 6)_rabbit = 18_rabbit.                                                        
2. Each operand has its own unit, but the two units can be related to produce one: One unit is a for a rate or ratio that acts on the unit of the second to produce a third unit from the rate unit. Examples:                                                                                                                                 ** 3_(rabbits/cage) x 6_cage = 3_(rabbits/cage x cage) x 6 = 3_rabbit x 6 = (3x6)_rabbit = 18_rabbit,                                                                                                                 ** 3_(miles/hour) x 6_hour = 3_(miles/hour x hour) x 6 = 18_mile.                                                 
3. Each operand has its own unit, but now multiplication ties them together. In the previous two cases the product is the number of "thing"s. Now the product is the number of pairings of one operand's unit with the other operand's unit. The meaning of the pairing varies with the problem. Here multiplication acts on the "things" themselves and not just on the number of things; unit sense is central to the problem; e.g. two perpendicular lengths, one dimensional figures, can be multiplied to produce a two dimensional, rectangular-shaped surface. This is not elementary mathematics, but the sense of it is. Now we have a multiplication that goes beyond multiple adding. Examples:                                                                                                          **  6_cage x  3_rabbit = (6 x 3)_(cage,rabbit) = 18_(cage,rabbit) , is a pairing of a cage unit with a rabbit unit. In this case 1 rabbit with 1 cage. Each of the 18 pairings can not exist at the same time, there are not enough rabbits, or there too many cages. There are 18 distinct ways to pair a cage with a rabbit. If we think of a rabbit as well as a cage as points, then a (cage,rabbit) is a pair of points that locates the pair. This is much like a two dimensional Cartesian coordinate system, and indeed the pair is the result of a Cartesian product of sets, in this case one of cages and another of rabbits.                                                                                                                                                                                                                                                           Let me end with an example that shows multiplication producing a measure of  a two-dimensional object, the area of a rectangle, from two one-dimensional objects, the Length side, 6_ftL, and the Width side, 3_ftW. The Length side is perpendicular to the Width side. Each side length has the measure unit, foot(ft). The area is given by                                                        **  6_ftL x 3ftW = (6 x 3) x( 1_ftL x 1_ftW) = 18_square-ft.                                                             

Summary 

1. You can construct the Natural Numbers you need, including a Natural Number count of Things.
2. On the system of these numbers, the Natural Number system, you can define a process for addition and one for multiplication that constructs a Natural Number for the sum or the  product. This also works for Natural Number counts of "Things". In later posts, the Integers, the positive and negative whole numbers, will be defined in terms of the Natural Number system. Integer arithmetic, including subtraction is  also defined in terms of Natural Number arithmetic. Finally the Rational Number system along with division is defined in terms of the Integer Number system; and now we have the complete picture of elementary school arithmetic based on the Natural Number system.
3. Multiplication can be multiple addition for either Natural Numbers of Natural Numbers of Things. It can also be a Cartesian Product that produces pairings. 

Conclusion

There is a straight forward constructive path through school arithmetic based on Natural Numbers.

DRAFT Natural Number Arithmetic

I try to construct a rigorous and relatively simple path from natural numbers to arithmetic for rational numbers. It is pieced together from Wiki articles on the natural, integer, and rational number systems. I would like to think it shows the existence of an accessible, and rigorous basis for all of school arithmetic, accessible for teachers to enrich and inform their instruction.

[I did not intend the font change. In published form it does not follow the word processing instructions.]

Rote learning can be put in its proper place: it can make simple, practical calculation possible. But at the end of the day we can rest assured there is firm support for what we are doing A little exposure to foundation plants the seed to grow mathematical sense; and besides that, seeing how thought leads to understanding is the unique pleasure of mathematics.

I am not qualified to present material in a ready-to-teach-young-students form. I do try to get at the roots of elementary arithmetic. What is presented is certainly sufficiently straight-forward to be inform teachers on the foundation of the material they teach. This material should put teachers in position to evaluate for themselves claims like "multiplication is really addition". It looks to me like arithmetic proceeds in a straight line from a foundation well within the grasp of young students, and certainly well within the grasp of their teachers.

The mathematical approach can only be an abstraction of what? To use the result of that abstraction it is not necessary to know "what" it is, or could be. Teaching mathematics to beginners, on the other hand, requires taking a stand, conscientiously or not. Since I am trying to teach myself, I eventually had to face up to having no plausible story for how the bootstrap could have started for me. This is not cognitive science, it is simply what I see as a possibility when I look at the mathematics: what I could have been doing before I started learning school mathematics?

Here is the story I imagine. I realized one thing came after another thing, it wasn't all one long hum. I heard sound from my most significant other; it came in bursts, one after the other, sometimes little bursts inside big bursts. Eventually I got the hang of this after, after, ... business. At the same time I noticed the leg going out in front of me followed my other leg going out out in front of me, sort of like one sound burst followed another. "after" was like "following". My feet made marks in the sand, one after the other. When I took a step from a rock there was a mark in the sand, when I continued walking more marks appeared. Eventually I came up with a song to hum when I was walking with a word for the first step followed by another word for this next step. I noticed the tree close to the to the rock was ug1-ug2-ug3-ug4-ug5, that is an ug5 walk from the rock.  

Counting

Now I would describe counting as the attribution of "1 more", call it "+1" to the count of the pile before I put a ---- onto it. It is the same pile that was empty, call its count "0"(zero), before I put a --- onto it: Counting produces a sequence of counts:

• starting with:          old-count = 0 
• proceeding with:    new-count = old-count +1
•                                old-count   = new-count
• and continuing       new-count = old-count +1
• 
  
• 
  
•  

Counting evaluates the sequential growth/augmentation of the pile in the process of adding a --- to the pile that was empty to begin with. It is a sequential process, it does carry the idea of progression, of "step after step ..." but it also carries a record of augmentation, the pile is getting bigger. We attend to the progression part, or rather we let Peano and his axioms for the sequence of Natural Numbers take care of it. There is a problem with calling them numbers at this stage because Peano's axioms alone are only requirements that a sequence of words must satisfy in order for

• the sequence to start with a unique word,  
• for each word in the sequence to be different from all other words and for each word to have a unique place in the sequence, and 
• for there to be a simple way to prove something for every word in the sequence.

The set of all the words is written as N = {0,1,2,3, ...}. Each one of the words is called a natural number, but at this stage we need to be careful to remember they are only what we say about them, and we say they are words that Peano calls natural numbers. Peano says:

• 0 is a natural number.
• Every natural number has a successor.
• 0 is not the successor of any natural number.
• if the successor of x equals the successor of y. then x equals y.
• if: [a] something (since we are talking about words that "something is a statement) is true about 0, and [b] if the truth of that statement for a number implies (means) it is true for the successor of that number, then [c] the statement is true for every natural number.                                                                  

Here is a place you have to be careful not to assume "1" or "2" mean what you remember they meant before we decided to get at the truth of things. "2" is not the successor of "1" is not the successor of 0 until we show how Peano's axioms let that be so. We have to prove what we say so we can know, really know the truth of what we are talking about. We are talking about natural number arithmetic. Is it true, and exactly what is it?

First we need to define addition. We do not know what is the successor of what. All we know is 

• that special word 0 is not the successor of any natural number, 0 is a natural number, and therefore 0 is not a successor of itself.
• Every natural number has a successor. 

If we use the usual numerals in the following, we are likely to use the properties we have come to associate with them in the proof that all those properties follow from just the Peano's axioms. For now we take the usual alphabet for the natural numbers. Use "+" for addition, then we define "+" by

• a + the-succ-of b = the-succ-of ( a + b), and
• a + 0 = a.

First notice that we are trying to define + on the left hand side of = by an expression on the rhs that contains + ;  rather like the definition of a word including the word itself. It works in this case because the b is the-succ-of c in a + b , and then  a + b = a + the-succ-of c = the succ-of (a + c), and we continue in this "recursive" manner until we inevitably arrive at a + 0. 0 is not the successor of an natural number. That is why setting a + 0 = a is part of the definition of addition. And finally we have + on the lhs but not on the right hand side, and the recursion stops. This is a general definition, it works for all natural numbers a and b, but recursion is slippery; we will provide some numerical calculation later to show how it all hangs together

We are free to name the-succ-of 0: let's say 1 := the-succ-of 0. 

• b + 1 = b + the-succ-of 0 = the-succ-of ( b + 0 ) = the-succ-of b,

b + 1 is the successor of b, where b can be any natural number. Using only: [a] 1 defined as the succ of 0, and [b] the defined addition for natural numbers, we can name/define them in accord with our expectations for them:

• 0 is not the successor of any natural number
• 1 := the-succ-of 0 ,        definition of 1
• 2 := the-succ-of 1 ,        definition of 2
• 3 := the-succ-of2 ,        definition of 3
• 4 := the-succ-of 3 ,        definition of 4
• 5 := the-succ-of 4 ,        definition of 5
• 6 := the-succ-of 5 ,        definition of 6
• 7 ...
• 8 ...
• 9 ...
• ...

From here on we will denote "the-succ-of" by "succ". The defined addition gives:

• 6 + 3 = 
• 6 + succ (2)  =  succ (6 +2) =  
• succ ( 6 + succ (1)) = succ(succ(6 + 1))  =
• succ(succ(6 + succ(0)) = succ(succ(succ(6 + 0))) = succ(succ(succ(6))) =
• succ(succ(7)) = succ(8) = 
• 9

This is a general pattern for recursion; construct the "program" to produce the result as a sequence of steps, for addition each step is a "succ"; until it comes to a step that can be executed; and then it runs back up the program, executing each step to produce the input for the next until it runs out of steps. Also note the associated "shape" of the program: the width starts out small, reaches a peak, and returns to small corresponding to first building out a counting program and then consuming it as each counting step is executed. 

How is this useful?

• It shows how mathematics works: it proves things about words that are specified only by simple relations among them. For the set of natural numbers it shows how the definition of an addition of two of these words/numbers leads to a relation between them that can be expressed with the numbers themselves.
• It shows how addition of natural numbers is true: it is the rigorous working out of the implications of a small number of requirements for the set of all the natural numbers.
• It displays the elegance and economy of mathematics. 
• It will turn out that all of school analysis can be related back to natural number arithmetic.  

Multiplication

Given that addition has been defined, multiplication can be defined in an analogous fashion by setting

• a x 0 = 0, and
• a x succ b = (a x b) + a.

For the experience represent the set of natural numbers as N = {0, #, ##, ###,  ...} with # = succ 0, etc. Then

• ###### x ### 
• (###### x succ ##)  = (###### x ##)  + ###### =
• (###### x succ #)  + ######  =  ((###### x #) + ######) + ###### =
• (( ###### + ######) + ######) =
• ( ############ + ###### ) =
•  ##################

We used

• (###### x #) = (###### x 0) + # = 0 +  ###### = ###### .

Sorry for any problems reading this. Actually there is an advantage in being limited to a  small number of symbols that can only be displayed in a straight line, but the advantage is mine; I am allowed to look for a scheme that makes sense without facing diversion by pretty notation. It is not good for you though.

Eventually I noticed word processing a multiplication of natural numbers in the # notation was mostly a matter of copying and pasting. And then it occurred to me that was true because the # notation for natural numbers is a self representation; e.g. pasting a # on to the end of natural number constructs its successor, and that leads to the realization a natural number can be representing by the thing that constructs it: a string of successor "operators" we have denoted by "succ" that extends to the left ( this is  just the way readers of mathematics are used to seeing it) and originates with a 0 on the right. It is best to think of succ as a small machine that pulls in the natural number on its right, makes the successor of that natural number, and puts it directly under itself, and then goes away. Starting from the right it is then easy to see that 6 can be represented by the construction

• 6 := succ succ succ succ succ succ 0

In the # representation of natural numbers: we can have

• for # = succ 0, 
• # + ##...  = ###... for any natural number; i.e,
• succ ##... := prepend # to ##...

We can go back and forth between the mathematical construction of a natural number with a string of succ operators, and a displayed representation of them made by executing a string of prepend instructions. In either case we could write a program to produce

• natural numbers as prepend(prepend( prepend( ... prepend(# to 0) ...))). There are n prepends to create an n length string of #s.
• n + m as prepend(m to n). where the natural numbers n and m are represented as n and m length strings of #s as shown in first bullet. The a correct program that implements the bullet above would produce a string of (n + m) #'s.
• n x m as prepend(prepend(...prepend(n to 0)...))). There are m prepends and again n is a natural number produced as the first bullet indicates. Now a correct program would generate a n x m length string of #s.

This is just an outline. You could work entirely in the # representation by implementing counting as picking a # off a string off the string of #'s that represents a natural number until there were no more. If this doesn't relieve you of some worries, forget it.

I did not start with the intent to show all natural number arithmetic, at bottom, is just fiddling with strings of #'s or succ's, by whatever names these may be known; it just turned out that way. All integer and rational number arithmetic can be expressed as natural number arithmetic, and hence all school arithmetic is, at bottom, just more of this fiddling. At the end of the day that is why computers can do it. What we learn in school is the syntax of working with the names of these strings. For computers this is called syntactic candy. For us, the time we must spend on syntax it is the price we pay for being slow calculators. Still I think we could write computer programs that showed young students how it all really works, we could call it the "The String Theory Game".

Windup 

What have we learned?

• There is a foundation for arithmetic.
• A little of how mathematics works.
• A good basis for going on to develop the foundation for the integer number system which incorporates negative integers, and the rational number system where fractions/ratios live. I hope to tackle this in the next post.

Currently there is a belief, or at least a strong wish, that natural number multiplication is really addition. This is the sort of issue that is difficult to address from the perspective of school arithmetic itself, where apparently a thing can get defined by the practices associated with the signs in it; if an expression only has plus signs it must be addition. The plus signs that appear in an intermediate stage of multiplication do reflect addition in the definition of multiplication; addition must be defined before multiplication can be defined, but the definitions of addition and multiplication are not the same: multiplication is not addition. For one thing thing, with a slight abuse of definition, addition counts by 1 from n, while multiplication counts by n from 0. There should be, at the least, someone available to every teacher who can explain questions at the level of: what is addition? what is multiplication? what is the difference?         

I left high school thinking math was mostly a matter of retrieval of small facts put in memory by rote learning and repetitive manipulation. This may have produced some number sense, but it is not number sense that underlies success in later work in mathematics. It is more a matter of mathematics sense. I came away with very little. Some exposure to the foundation of arithmetic would have been helpful.